
Assume \(S \rightarrow A\)  is not an isomorphism. Applying the assumption to the (PB)-cover \(\phi : S \rightarrow A\)  and to the (finitely generated) subgroup \(\Omega _A\) , we get a finite-index coset \(C \subseteq \Omega _A\)  with the property that, for every \(c \in C\) , the fibre \(\phi ^{-1}(c)\)  has no \(K\) -rational points. It follows immediately that \(C \times \Omega _B\)  is a finite-index coset of \(\Omega _A \times \Omega _B\)  such that, for any \((c,b) \in C \times \Omega _B\) , we have \(\pi ^{-1}((c,b))(K) \subseteq \psi ^{-1} \phi ^{-1}(c) (K) = \emptyset \) . Since \(\pi ^{-1}((c,b))\)  is reduced by assumption, this proves that \(\pi :Z\rightarrow A\times B\)  has property (PF) in this case.

Assume that \(S \rightarrow A\)  is an isomorphism, so that \(Z \rightarrow A\)  has geometrically connected fibres. Since \(K\)  is of characteristic zero and \(Z\)  is a normal geometrically integral variety over \(K\) , \(Z_{\overline{K}}\)  is normal by [1]}. Hence, the generic fibre of \(Z_{\overline{K}} \rightarrow {A}_{\overline{K}}\)  is normal and integral by a simple localization argument. Therefore, by [2]} and [3]}, there is a dense open subscheme \(\mathcal {U}\)  of \(A\)  over which the geometric fibres of \(Z \rightarrow A\)  are normal (integral) varieties. In particular, for every \(a \in \mathcal {U}(K)\) , the morphism \(Z_a \rightarrow \lbrace a\rbrace  \times B\)  is a cover of \(B\)  over \(K\) .
Let \(\eta _A \in A\)  be the generic point.
By Lemma REF , there is a finite étale cover \(\widetilde{B}\rightarrow B_{\eta _A}\)  and a cover \(Z_{\eta _A}\rightarrow \widetilde{B}\)  such that \(Z_{\eta _A}\rightarrow B_{\eta _A}\)  factors as
\(Z_{\eta _A} \rightarrow \widetilde{B} \rightarrow B_{\eta _A},\) 
and the morphism \(Z_{\overline{K(A)} }\rightarrow \widetilde{B}_{\overline{K(A)}}\)  is a (PB)-cover of the abelian variety \(\widetilde{B}_{\overline{K(A)}}\) . Let \(\tilde{Z}\)  be the normalization of \(A \times B\)  in (the function field of) \(\widetilde{B}\) . Note that \(\tilde{Z}\)  is a normal variety over \(K\)  and that the factorization REF  induces a factorization
\(Z \rightarrow \tilde{Z} \xrightarrow{} A \times B,\) 
with \(\tilde{Z}_{\eta _A}= \widetilde{B}\) .
By construction \(\lambda \)  is vertically ramified over \(A\)  (see Definition REF ). Moreover, \(\lambda \)  is a (PB)-cover of \(A\times B\) , as it is a factor of the (PB)-cover \(Z \rightarrow A \times B\) .

Suppose that \(\tilde{Z}=A \times B\)  (i.e., \(\deg \lambda =1\) ), so that \(Z_{\eta _A}\rightarrow B_{\eta _A}\)  is a (PB)-cover. Therefore, by applying Lemma REF  to the abelian scheme \(\mathcal {A} := A\times B\rightarrow A=:S\) , there is a dense open subscheme \(U\subset A\)  such that, for every \(a\)  in \(U(K)\) , the morphism \(Z_a \rightarrow \lbrace a\rbrace  \times B\)  is (PB)-cover of \(B\) .
Since \(\Omega _A\)  is dense in \(A\) , there is a \(K\) -rational point \(a\)  of \(U\)  contained in \( \Omega _A\) .
In particular, the assumption on \((B,\Omega _B)\)  implies that there is a finite-index coset \(C\)  of \(\Omega _B\)  such that, for every \(c\)  in \(C\) , the scheme \(\pi ^{-1}((a,c))\)  is integral. By Corollary REF , the existence of such a point implies that \((\pi , \Omega _A \times \Omega _B)\)  satisfies (PF) as desired.

Suppose that \(\deg \lambda \ge 2\) . Then, property (PF) for \((\pi , \Omega _A \times \Omega _B)\)  is implied by property (PF) for \((\lambda , \Omega _A \times \Omega _B)\)  (see Remark REF ). Thus, we may and do assume that \(\tilde{Z}=Z\) , so that \(\pi :Z \rightarrow A \times B\)  is vertically ramified over \(A\) . Applying Lemma REF  to \({\pi }_{\overline{K}}: Z_{\overline{K}} \rightarrow A_{\overline{K}} \times B_{\overline{K}}\)  yields the existence of a commutative diagram
\({& \overline{Z^{\prime }}\cong \overline{S^{\prime }} \times \overline{B^{\prime }} [r] [d] [ld] & Z_{\overline{K}} [d] \\\overline{S^{\prime }} [rd]^{\overline{\phi ^{\prime }}} & A_{\overline{K}} \times \overline{B^{\prime }} [r] [d] & A_{\overline{K}} \times B_{\overline{K}} & \hspace{-36.135pt}, \\& A_{\overline{K}}}\) 
where:

\(\overline{S^{\prime }}, \overline{B^{\prime }}, \overline{Z^{\prime }}\)  are normal varieties over \(\overline{K}\) ;

\(\overline{S^{\prime }} \rightarrow A_{\overline{K}}\)  is a ramified cover;

\(\overline{Z^{\prime }} \rightarrow Z_{\overline{K}}\)  and \(\phi _B:\overline{B^{\prime }} \rightarrow B_{\overline{K}} \)  are finite étale; in particular, we may fix a structure of abelian variety on \(\overline{B^{\prime }}\)  that makes \(\phi _B\)  an isogeny;

\(\overline{Z^{\prime }}\)  is a connected component of the fibred product \(Z_{\overline{K}} \times _{B_{\overline{K}}} \overline{B^{\prime }}\) . In particular, since \(Z_{\overline{K}} \times _{B_{\overline{K}}} \overline{B^{\prime }}\)  is connected by Lemma REF , the square is cartesian;

\(\overline{Z^{\prime }} \rightarrow \overline{S^{\prime }} \rightarrow A_{\overline{K}}\)  is the Stein factorization of \(\overline{Z^{\prime }} \rightarrow A_{\overline{K}}\) ; hence \(\overline{S^{\prime }} \rightarrow A_{\overline{K}}\)  is a (PB)-cover by Lemma REF .

We choose a finite extension \(L/K\)  over which the entire diagram (REF ) is defined. More precisely, we let \(S^{\prime }, Z^{\prime }\)  be varieties over \(L\)  and \(B^{\prime }\)  be an abelian variety over \(L\)  such that \(S^{\prime }_{\overline{K}}\cong \overline{S^{\prime }}\) , \(B^{\prime }_{\overline{K}}\cong \overline{B^{\prime }}\)  (as abelian varieties over \(\overline{K}\) ), and \(Z^{\prime }_{\overline{K}}\cong \overline{Z^{\prime }}\) . Moreover, let \(\phi ^{\prime }:S^{\prime } \rightarrow A_L\)  be a morphism such that \(\phi ^{\prime }_{\overline{K}}=\overline{\phi ^{\prime }}\) . Also, replacing \(L\)  by a finite extension if necessary, we have that \(\Omega _B \subset \phi _B(B^{\prime }(L))\) .
Since \(S^{\prime } \rightarrow A_L\)  is a non-trivial (PB)-cover of \(A_L\) , our assumption on \((A_L,\Omega _A)\)  implies that there is a finite-index coset \(C \subseteq \Omega _A \subset A_L(L)\)  such that, for every \(c \in C\) , the scheme \((\phi ^{\prime })^{-1}(c)\)  is integral. In particular, as the degree of \(\phi ^{\prime }\)  is at least two, the set \(((\phi ^{\prime })^{-1}(c))(L)\)  is empty. Thus, for every \((c,b)\)  in the finite index coset \( C\times \Omega _B\)  of \(\Omega _A\times \Omega _B\) , we have that \( \pi _L^{-1}((c,b))(L)\)  (hence \(\pi ^{-1}((c,b))(K)\) ) is empty. It follows that \(\pi \)  satisfies (PF) as desired.


